Streaming PCA and Subspace Tracking: The Missing Data Case

TL;DR

This paper reviews algorithms for streaming PCA and subspace tracking with missing data, emphasizing low complexity, convergence guarantees, and benchmarking in big data scenarios.

Contribution

It provides a comprehensive survey of classical and recent algorithms, highlighting their algebraic and geometric insights, and evaluates their performance with missing data.

Findings

Algorithms can be adapted for missing data scenarios.

Convergence guarantees are established for various methods.

Benchmark results compare performance under different conditions.

Abstract

For many modern applications in science and engineering, data are collected in a streaming fashion carrying time-varying information, and practitioners need to process them with a limited amount of memory and computational resources in a timely manner for decision making. This often is coupled with the missing data problem, such that only a small fraction of data attributes are observed. These complications impose significant, and unconventional, constraints on the problem of streaming Principal Component Analysis (PCA) and subspace tracking, which is an essential building block for many inference tasks in signal processing and machine learning. This survey article reviews a variety of classical and recent algorithms for solving this problem with low computational and memory complexities, particularly those applicable in the big data regime with missing data. We illustrate that streaming PCA and subspace tracking algorithms can be understood through algebraic and geometric perspectives, and they need to be adjusted carefully to handle missing data. Both asymptotic and non-asymptotic convergence guarantees are reviewed. Finally, we benchmark the performance of several competitive algorithms in the presence of missing data for both well-conditioned and ill-conditioned systems.